Current Search: Categories Mathematics (x)
View All Items
 Title
 Weakly integrally closed domains and forbidden patterns.
 Creator
 Hopkins, Mary E., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

An integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally...
Show moreAn integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every finite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/FAU/199327
 Subject Headings
 Mathematical analysis, Algebra, Homological, Monoids, Categories (Mathematics), Semigroup algebras
 Format
 Document (PDF)
 Title
 Rings of integervalued polynomials and derivatives.
 Creator
 Villanueva, Yuri., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integervalued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D)  {f e K [X]lf(k) (E) c...
Show moreFor D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integervalued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D)  {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, onedimensional, analytically irreducible, with finite residue field.
Show less  Date Issued
 2012
 PURL
 http://purl.flvc.org/FAU/3356899
 Subject Headings
 Rings of integers, Ideals (Algebra), Polynomials, Arithmetic algebraic geometry, Categories (Mathematics), Commutative algebra
 Format
 Document (PDF)
 Title
 Integervalued polynomials and pullbacks of arithmetical rings.
 Creator
 Boynton, Jason, Florida Atlantic University, Klingler, Lee
 Abstract/Description

Let D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the ngenerator property for D is equivalent to the ngenerator property for Int(E, D), which is equivalent to strong (n + 1)generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient...
Show moreLet D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the ngenerator property for D is equivalent to the ngenerator property for Int(E, D), which is equivalent to strong (n + 1)generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Prufer domains R between D[X] and K[X]such that the conductor C of K[X] into R is nonzero. As an application, we show that for n > 2, such a ring R has the ngenerator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
Show less  Date Issued
 2006
 PURL
 http://purl.flvc.org/fcla/dt/12221
 Subject Headings
 Polynomials, Ideals (Algebra), Rings of integers, Categories (Mathematics), Arithmetical algebraic geometry
 Format
 Document (PDF)